Overview

Students connect polynomial arithmetic to computations with whole numbers and integers. Students learn that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. This unit helps students see connections between solutions to polynomial equations, zeros of polynomials, and graphs of polynomial functions. Polynomial equations are solved over the set of complex numbers, leading to a beginning understanding of the fundamental theorem of algebra. Application and modeling problems connect multiple representations and include both real world and purely mathematical situations.

The student materials consist of the student pages for each lesson in Module 1.

The copy ready materials are a collection of the module assessments, lesson exit tickets and fluency exercises from the teacher materials.

Week 1: September 5

Notes:

Lesson 1: Students write polynomial expressions for sequences by examining successive differences. They are engaged in a lively lesson that emphasizes thinking and reasoning about numbers and patterns and equations.

Lesson 2: Students use a variation of the area model referred to as the tabular method to represent polynomial multiplication and connect that method back to application of the distributive property.

Week 2: September 11

Notes:

Lesson 3: Students continue using the tabular method and analogies to the system of integers to explore division of polynomials as a missing factor problem. In this lesson, students also take time to reflect on and arrive at generalizations for questions such as how to predict the degree of the resulting sum when adding two polynomials.

Lesson 4: Students are ready to ask and answer whether long division can work with polynomials too and how it compares with the tabular method of finding the missing factor.

Lesson 5: This lesson gives students additional practice on all operations with polynomials and offers an opportunity to examine the structure of expressions such as recognizing that (n(n+1)(2n+1))/6 is a 3rd degree polynomial expression with leading coefficient 1/3 without having to expand it out.

Week 3: September 18

Notes:

Lesson 6: Students extend their facility with dividing polynomials by exploring a more generic case; rather than dividing by a factor such as (x+3), they divide by the factor (x+a) or (x-a).

Lesson 7: The work from the previous lesson gives students the opportunity to discover the structure of special products such as (x-a)(x^2+ax+ a^2) in Lesson 7 and go on to use those products in Lessons 8–10 to employ the power of algebra over the calculator.

Lesson 8: Students find they can use special products to uncover mental math strategies and answer questions such as whether or not 2^100-1 is prime.

Lesson 9: Students consider how these properties apply to expressions that contain square roots.

Week 4: September 25

Standards Addressed in Week's Lessons

Topic B:

N-Q.A.2

A-SSE.A.2

A-APR.B.2

A-APR.B.3

A-APR.D.6

F-IF.C.7

Notes:

Lesson 10: Students use special products to find Pythagorean triples.

Lesson 11: The topic culminates with Lesson 11 and the recognition of the benefits of factoring and the special role of zero as a means for solving polynomial equations.

Lessons 12–13: Students are presented with the first obstacle to solving equations successfully. While dividing a polynomial by a given

factor to find a missing factor is easily accessible, factoring without knowing one of the factors is challenging.

Week 5: October 2

Standards Addressed in Week's Lessons

Topic B:

N-Q.A.2

A-SSE.A.2

A-APR.B.2

A-APR.B.3

A-APR.D.6

F-IF.C.7

Notes:

Lessons 14–15: Students find that another advantage to rewriting polynomial expressions in factored form is how easily a polynomial function written in this form can be graphed. Students read word problems to answer polynomial questions by examining key features of their graphs. They notice the relationship between the number of times a factor is repeated and the behavior of the graph at that zero.

Lessons 16–17: Students encounter a series of more serious modeling questions associated with polynomials, developing their fluency in translating between verbal, numeric, algebraic, and graphical thinking.

Week 6: October 10

Module

Topic

Notes:

Lessons 18–19: Students are presented with their second obstacle: “What if there is a remainder?” They learn the remainder theorem and apply it to further understand the connection between the factors and zeros of a polynomial and how this relates to the graph of a polynomial function.

OMIT Lessons 20 and 21: These lessons have been omitted because they reapply skills that were already taught and there is not enough time given pacing considerations.

Standards Addressed in Week's Lessons

Topic B:

N-Q.A.2

A-SSE.A.2

A-APR.B.2

A-APR.B.3

A-APR.D.6

F-IF.C.7

Mid-Module Assessment:

A-SSE.A.2

A-APR.C.4

N-Q.A.2

A-SSE.A.2

A-APR.B.2

A-APR.B.3

A-APR.D.6

F-IF.C.7

Week 7: October 16

Module

Topic

Notes:

Lesson 22: Students expand their understanding of the division of polynomial expressions to rewriting simple rational expressions (A-APR.D.6) in equivalent forms.

Lesson 23: Students learn techniques for comparing rational expressions numerically, graphically, and algebraically.

Lessons 24–25: Students learn to rewrite simple rational expressions by multiplying, dividing, adding, or subtracting two or more expressions. They begin to connect operations with rational numbers to operations on rational expressions.

Standards Addressed in Week's Lessons

Topic C:

A-APR.D.6

A-REI.A.1

A-REI.A.2

A-REI.B.4

A-REI.C.6

A-REI.C.7

G-GPE.A.2

Week 8: October 23

Module

Topic

Notes:

Lesson 26: The practice of rewriting rational expressions in equivalent forms in Lessons 22–25 is

carried over to solving rational equations in Lesson 26.

Lesson 27: This lesson expands the work of Lesson 26 to include working with word problems that require the use of rational equations.

Lessons 28–29: These lessons turn to radical equations. Students learn to look for extraneous solutions to these equations as they did for rational equations.

Standards Addressed in Week's Lessons

Topic C:

A-APR.D.6

A-REI.A.1

A-REI.A.2

A-REI.B.4

A-REI.C.6

A-REI.C.7

G-GPE.A.2

Week 9: October 30

Standards Addressed in Week's Lessons

Topic C:

A-APR.D.6

A-REI.A.1

A-REI.A.2

A-REI.B.4

A-REI.C.6

A-REI.C.7

G-GPE.A.2

Notes:

Lessons 30–32: Students solve and graph systems of equations including systems of one linear equation and one quadratic equation and systems of two quadratic equations.